Download A mechanistic classical laboratory situation violating the Bell

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Published as: Aerts, D., 1991, A mechanistic classical laboratory situation violating the Bell
inequalities with 2·√2, exactly 'in the same way' as its violations by the EPR experiments, Helv.
Phys. Acta, 64, 1-23.
A mechanistic classical laboratory situation violating the Bell inequalities
√ 2, exactly 'in the same way' as its violations by the EPR experiments.
with 2·√
Diederik Aerts,
FUND and CLEA,
Vrije Universiteit Brussel,
Krijgskundestraat 33, 1160, Brussels.
e-mail: [email protected]
1.
Abstract :
We present a macroscopical mechanistic classical laboratory situation, and a classical
macroscopical entity, and coincidence measurements on this entity, that lead to a violation of the
Bell inequalities corresponding to these coincidence measurements. The violation that we obtain
with these coincidence measurements is exactly the same as the violation of the Bell inequalities by
the well known coincidence measurements of the quantum entity of two spin 1/2 particles in a
singlet spin state. With this we mean that it gives rise to the same numerical values for the
expectation values and the same numerical value 2·√2 for the expression used in the Bell
inequality. We analyze the origin of the violation, and can formulate the main difference between
the violation of Bell inequalities by means of classical entities and the violation of Bell inequalities
by means of quantum entities. The making clear of this difference can help us to understand better
what the quantum-violation could mean for the nature of reality. We think that some classical
concepts will have to be changed, and new concepts will have to be introduced, to be able to
understand the reality of the quantum world.
2.
Introduction.
The violation of Bell inequalities by measurements on quantum mechanical entities is
certainly one of the most stimulating happenings for the research on the physical meaning of the
quantum formalism for the nature of reality. A lot of reflections have been made by many
physicists and philosophers after the finding of Bell
1)
that there was something 'really mysteri-
ous' about the structure of the correlations predicted by quantum mechanics for coincidence spindirection measurements on a quantum entity consisting of two spin 1/2 particles in a singlet spin
state. Meanwhile the structure of these correlations has been confirmed by measurements, which
shows that the mystery is in the nature of the reality of these quantum entities.
2
Many profound and careful reasonings have been made about the possible significations for
2,3,4,5)
the nature of reality of the appearance of the structure of these correlations
of agreement has been reached among the physicists working in the field
but not any kind
6)
.
A problem with 'theoretical reasonings' on such fundamental subjects, is that often unconsciously one uses hypothesis that are more or less hidden in the mathematics that goes along
with the theoretical reasoning. These hypothesis seem perhaps obviously satisfied for the 'physical
picture' that one has in mind of the situation in question, but are perhaps in general not so
obviously satisfied in reality. With this principle idea in mind "Reality can be more complicated
than one imagines at the first place", already some time ago we gave an example of a
macroscopical physical entity that could violate the Bell inequalities
7,8)
. We have improved the
original example, and presented and analyzed it in different ways, with the aim of trying to understand more about the physical meaning of the violation of Bell inequalities by quantum entities for
the nature of reality
9,10,11,12)
. Our main aim was to present a macroscopical example violating
Bell inequalities in the ordinary macroscopical reality, with the purpose to see "how near one can
succeed in producing this strange structure of correlations in ordinary reality". Meanwhile we also
have analyzed in detail what are the sometimes 'additionally unconsciously used assumptions' (we
have called them AUUA) in the different derivations of the Bell (type) inequalities
13)
.
In the analysis of this example we have been considering the following form of the Bell
inequality, originally derived by Bell in
14)
:
| E(a,b ) − E(a,b') | + | E(a',b) + E(a',b') |
≤
2
(1)
where E(a,b) is the expectation value of the observable corresponding to a coincidence
measurement m(a,b) on a entity S, where for outcomes {yes,yes} and {no,no} of the coincidence
measurement we agree to give value +1 to the corresponding observable, and for outcomes
{yes,no} and {no,yes} we agree to give value -1 to this observable. The same definition holds for
the other quantities E(a,b'), E(a',b), and E(a',b').
These quantities where introduced by Bell, having in mind the quantum entity consisting of
two spin 1/2 particles, produced in a singlet spin state. The spin part of the state vector of this
entity in quantum mechanics is represented by the vector :
Ψ
S
=
1
√2
[u
n
⊗ u
−n
− u
−n
⊗ u ]
n
(2)
where n is a normalized vector in three dimensional Euclidean space, and un is the corresponding
vector in the two dimensional complex Hilbert space that is used in quantum mechanics to describe
the spin of a spin 1/2 particle. If n = (cosφ·sinθ, sinφ·sinθ, cosθ) then we can take un =
3
(e
-iφ/2
·cos θ /2, e
iφ/2
·sin θ /2). Remark that -n = (cos(π+φ) ·sin( π−θ) , sin(π+φ) ·sin( π−θ) , cos(π−θ ))
and hence u-n = (-i·e
-iφ/2
·sinθ/2, i·e
iφ/2
·cosθ/2). We have σ·n un = un and σ·n u-n = -u-n, such that
un and u-n describe the states in which a particle has respectively spin 'up' and spin 'down' along
the direction n. The singlet state is 'spherically' symmetric, n can be any direction in space, (2)
always leads to the same vector ΨS.
The measurements considered by Bell are the ones originally proposed by Bohm
15)
in his
reasoning on the EPR problem, and consist of making coincidence measurements of the spin on
this entity consisting of two spin 1/2 particles in the singlet spin state along well defined directions
of space.
If we calculate the expectation value E(a,b) of the observable corresponding to the coincidence measurement m(a,b) following the rules of quantum mechanics, we find
E (a,b) = < Ψ
S
| σ·a ⊗ σ·b | Ψ
S
> = − a·b
(3)
It is very easy to see that for a good choice of the space directions a, b, a', b', the Bell inequalities are violated for these quantum mechanical expectation values. For example consider the situation as shown in (fig 1) then,
a
The choice of space directions to measure
b
the spin, that allows a maximum violation of
the Bell inequalities (fig 1).
a'
b'
we find :
| E( a,b ) − E( a,b' ) | + | E( a',b ) + E( a',b' ) | = 2·√ 2
(4)
which violates the inequality. This is the maximum violation that quantum mechanics predicts for
the inequalities. In the case of the macroscopical example that violates the inequalities, presented in
7,-,13)
the violation is not 2·√2 as in the quantum case, but 4, which is more than the quantum
violation. It seems at first sight a little bit strange that we can violate the Bell inequalities more with
coincidence measurements on a classical macroscopical entity, than with coincidence
measurements on the quantum entity. We shall understand why this is the case after the analysis
that we will present in this paper. We shall see that in a certain sense the presence of the quantum
probabilities temper the violation of the Bell inequalities.
In this paper we want to present a macroscopical entity, violating the Bell inequalities, 'exactly in the same way' as the quantum entity of two spin 1/2 particles in the singlet spin state do.
With 'exactly in the same way', we mean, as to the numerical value, but also as to the values of
4
the expectation values of the different correlations. We could use the quantum formalism in its
details to give a description of the macroscopical entity that we will present in this paper. This does
of course not mean that we want to pretend that this is the way things happen with the spin entity.
Not at all. Our aim is to limit of strictly the classical ways of violations of the inequalities, not by
considering theoretical reasonings, but by looking at explicit examples of real physical entities in
the ordinary reality.
In the second section we will present again the old example, because of its simplicity,
together with the reasoning that tries to find out the AUUA, in the derivation of the Bell
inequalities. In the third section we will present a classical spin-model, that we need to built our
macroscopical entity that 'imitates' the violation of the inequalities by means of the quantum entity
consisting of two particles in the singlet spin state. This classical spin-model has been presented
earlier in
16,17)
. In the fifth section we will present the example. In the sixth section we will
analyze the example, and in the seventh section we will speculate about the possible meaning of it
all. We want to repeat again that it is our main intention on the hand of these examples to see how
far we can go, without giving up profound principles about the nature of reality, in creating
situations that realize this strange form of correlations that lead to the violation of Bell inequalities.
3.
The classical macroscopical example that violates Bell inequalities in
classical reality.
The entity consists of two vessels V1 and V2 that contain each 10 litre of water and are
connected by a tube. The measurement m(a) consists of taking the water out of V1 with a siphon
and collecting it in a reference vessel R1. If we collect more than 10 litres, the outcome for m(a) is
'yes', and if we collect less than 10 litre the outcome for m(a) is 'no'. The measurement m(b) is
the same as m(a) but performed on V2. The coincidence measurement m(a,b) consists of
performing m(a) and m(b) together. This coincidence measurement creates correlations. Indeed, if
we find more than 10 litres in R1, then we find less than 10 litres in R2, and vice verse. The
correlations are detected at both sides when the water stops flowing, this means simultaneously.
Hence the events that correspond to the detection of the correlations are space-like separated
events.To calculate Bell inequalities we have to introduce two other measurements.
The measurement m(a') consists of taking 1 litre of water out of V1 and checking whether
the water is transparent.
5
m(a)
V
V
1
m(b)
2
A classical macroscopical
system that violates Bellinequalities (fig 2).
R
R
2
1
If the water is transparent , then the outcome of m(a') is 'yes', and if it is not transparent the
outcome is 'no'. The measurement m(b') is the same as m(a') but performed on V2. We make
coincidence measurements m(a,b'), m(a',b) and m(a',b'). We now define the following random
variables : E(a) = +1 if m(a) gives 'yes', and E(a) = -1 if m(a) gives 'no'. In the same manner we
define E(b), E(a'), and E(b'). We also define the random variables for the coincidence experiments,
E(a,b) = +1 if m(a,b) gives "yes,yes" or "no,no", and E(a,b) = -1 if m(a,b) gives
"yes,no" or "no,yes".
If the entity is in such a state that the two vessels of water contain 10 litre of transparent
water, then
E(a,b) = -1 , E(a',b) = +1 , E(a,b') = +1 , E(a',b') = +1. Hence :
| E(a,b) - E(a,b') |
+
| E(a',b) + E(a',b') | = +4
> +2.
(5)
This shows that Bell inequalities are violated, and if we compare with the quantum mechanical violation, we see that our entity violates the Bell inequalities 'more' than the quantum mechanical entity of the two spin 1/2 particles in the singlet spin state. If we present our second
classical example in section 4, we will understand why this is the case.
In relation with this example we will now analyze the original derivation of the Bell
inequalities using a locality hypothesis for the case of deterministic hidden variables. Later on the
inequalities have been derived for the case of non-deterministic hidden variables, but since in any
way in this macroscopical example the hidden variables are deterministic, we need not consider
these generalizations at this moment. We will see that also this locality hypothesis is violated for
the case of our example.
A deterministic hidden variable theory is a theory that postulates the existence of states of the
entity such that all observables have a determined outcome when the state is known. Let us denote
by Γ the set of these states λ. Hence in such a theory, E(a,b) has a determined outcome E(a,b, λ)
for every state λ. Bell introduces then the following hypothesis :
Bell locality hypothesis : For all measurements m(a), m(b), and m(a,b) and for all λ we have:
E(a,b, λ ) = E(a, λ )·E(b, λ )
(6)
6
The physical meaning of this hypothesis is that the result of the measurement m(a) only depends
on the state λ and not on the measurement m(b). Since this Bell locality hypothesis implies Bell
inequalities to be satisfied, our example must also violate the Bell locality hypothesis. Let us try to
see why this is so. It is very easy to specify the deterministic hidden variables for our entity.
Indeed, if we specify for example the diameters λ1 and λ2 of the two siphons, the outcomes of all
the measurements are determined. Hence we can write :
E(a,b, λ 1 , λ2 ) = E(a, λ 1 , λ2 )·E(b, λ 1 , λ2 ),
(7)
E(a, λ1, λ2 ) = +1 and E(b, λ1, λ2) = -1 if
λ 1 > λ2
(8)
E(a, λ1, λ2 ) = -1 and E(b, λ1, λ2) = +1 if
λ 1 < λ2
(9)
This is a correct factorisation if one performs the coincidence measurement m(a,b). If one wants
however to use the same E(a, λ1, λ2 ) to factorize the random variable E(a,b') from the coincidence
measurement m(a,b') it does not work any more. Indeed, m(a) performed together with m(b') always gives the outcome 'yes'. This means that the value of E(a) does not only depends on the
states λ1, λ2 but also on the fact that we perform measurement m(b) or m(b'), and
E(a, λ 1, λ2, b ) ≠ E(a, λ 1, λ2, b' )
(10)
since E(a, λ1, λ2, b ) = +1 if λ1 > λ2 and E(a, λ1, λ2, b ) = -1 if λ1 < λ2 while E(a, λ1, λ2, b') = +1
for all λ1, λ2.
Bell has put forward this locality hypothesis having in mind the entity consisting of two
spin-1/2 particles in the singlet spin state. Why do people find this locality hypothesis 'natural' for
this entity? Because they imagine the entity to be an entity consisting of two spin-1/2 particles
located in different widely separated regions of space while they are flying apart. And indeed, for
two entities located in widely separated regions of space, with no connection between them, the
Bell locality hypothesis seems to be a natural hypothesis to be satisfied. But for two entities that
actually form a whole (like the water in the two vessels) it is very easy to violate the Bell locality
hypothesis, and hence also the Bell inequalities.
We would now like to find out the physical reason for the violation of the Bell inequalities
by our macroscopical entity. We can already understand very much if we consider the nature of the
hidden variables λ1 and λ2. These are not hidden variables of the state of the entity 'water', before
the measurement, because the state of the water (and hence the reality of the water) is completely
determined by the fact that the volume is 20 litre. And λ1 ( λ2 ) is a hidden variable of the
measurement m(a) ( m(b) ), but not of the measurement m(a') ( m(b') ). Hence if we would
analyze this situation in the scheme of 'non classical probability models', as we have done in
16,17)
, we would classify the hidden variables λ1 and λ2 as representing 'hidden measurements',
7
and not 'hidden states', and as is shown in
18)
, the water as in the example has a 'non classical'
probability model. This explains from a probabilistic point of view why we can violate Bell
inequalities with our example of the vessels of water. Indeed, the correlations that are detected by
the measurement m(a,b) were not present before, but are created during the measurement, and
therefore they can violate the Bell locality hypothesis. We propose to call correlations that were not
present before the measurement and are created by and during the measurement 'correlations of
the second kind'. Correlations that were already present before the measurement and are only
detected by the measurement, we will call 'correlations of the first kind'.
Let us give an example of such correlations of the first kind. Consider a entity consisting of
two material point particles moving in space and having total momentum zero. A coincidence
measurement of the momenta of the individual particles gives us correlated results. These correlations were however already present before the coincidence measurement. The measurement only
detects the correlations and does not create them. These kinds of correlations can never be used to
violate Bell inequalities, because the result of a measurement on one of the particles will never
depend on what measurement is being performed on the other particle. If we read the paper on the
history of the EPR paper of Max Jammer in
19)
, it becomes clear that this difference was exactly
the point that puzzled Einstein, and was at the origin of the EPR article.
Let us try to summarize : If we consider correlations that are created by and during the
coincidence measurement m(a,b) (correlations of the second kind), then it is possible to violate
Bell inequalities and the Bell locality hypothesis by means of this coincidence measurement and
some other measurement, because the outcome of measurement m(a) will in general depend on
whether we perform m(a) together with m(b) or with some other measurement m(b'). If we
consider correlations that were already present before the coincidence measurement, then the Bell
locality hypothesis will be satisfied, and Bell inequalities cannot be violated.
Statement : The possibility of violating Bell inequalities is not only a property of quantumentities. Bell inequalities can also be violated by coincidence measurements on a classical
macroscopical entity. In fact Bell inequalities can always be violated if during the coincidence experiments one breaks one entity into separated pieces, and by this act creates the correlations. In
analogy with the example of the vessels of water, a lot of other macroscopical entities violating
Bell inequalities can be invented. But there is of course no mystery in these violations, because we
see with our own eyes, inside our own human reality what happens.
Let us now introduce a macroscopical model for the spin of a spin 1/2 particle, and then
construct the macroscopical example that 'imitates' the quantum way of violating the Bell
inequalities.
8
4.
A macroscopical classical model for the spin of a quantum particle.
The classical macroscopical spin model that we will present in this section has been pre16,17)
sented in
with the aim of giving a possible explanation for the the non-classical character of
the quantum probability model. It is shown in
16,17)
that a lack of knowledge about the measure-
ments on a physical entity gives rise to a non classical probability calculus for this physical entity.
It is also shown that the non classical probability calculus of quantum mechanics can be interpreted
as being the result of a lack of knowledge about the measurements. It is as a specific example of
such a classical model giving rise to a quantum mechanical probability calculus, that the spin
model that we will present now, is introduced in
16,17)
. In
16)
we have constructed an example by
using charges that can move under the influence of the Coulomb force on the surface of a sphere.
In
17)
we have considered masses that move under the influence of the gravitational force. We
want to point out here, that the aspect of using charges or masses, makes the example physically
real, but is not the most important part of it. The important point is that we can present a
'mechanistic' example, constituted of particles that move under influence of interactions in our
ordinary three dimensional Euclidean space. Hence in the presentation of the example here, we
will put full attention to this aspect. Let us give the model.
The classical macroscopical system that we consider is a particle characterized by a parameter
q (in
16)
this parameter q was taken to be a fixed positive charge, in
17)
it was taken to be a fixed
mass, but this is of no essential importance for the mechanistic aspect of the example). We will
from now on indicate this particle by the parameter q, and speak of the particle q.
We give a detailed description of the measuring apparatus and the measurements. We have a
rigid rod of a certain length l (see fig 3). At the end-points of the rod are two particles. One particle
is characterized by a parameter q
1
, and we will call it particle q , and the other particle is
characterized by a parameter Q - q
= q
1
(negative charges in
particle q
1
16)
2
and masses in
1
where Q is a fixed parameter and we will call it particle q
2
17)
). The rigid rod is placed fixed in the laboratory such that
is in space direction a, and particle q
2
in space direction -a in a plane orthogonal to
some fixed direction x. The particle q can be put in the neighbourhood of the measurement
apparatus, and will then be attracted by the particle q
1
and q
2
of the measuring apparatus.
We suppose that this happens in a viscous medium, such that under the influence of friction,
finally the particle q will end up at the place of one of the particles q
apparatus. If it ends up at particle q
1
1
or q
2
of the measuring
we give the outcome "a-up", and if it ends up with particle q
we give the outcome "a-down" for the measurement m(a).
2
9
direction a
particle 1
q
F
1
direction b
The measuring apparatus consists of a rigid rod of
q
1
length l with at the endpoints two particles, particle q
2
F
the particle q move under the influence of the forces of
2
l/2
direction x
rigid rod of length l
particle 2
q
1
and particle q . The measurement m(a) consists of letting
attraction and be captured by one of the two particles of
the measurement apparatus. If it is captured by particle q
then we give outcome "a-up" to measurement m(a), if
it is captured by particle q
2
1
we give outcome "a-down"
to the measurement m(a) (fig 3).
2
direction -a
We can now start making repeated measurements with measurement m(a) on particle q in a state
which is determined by a direction b and length l/2 from the centre of the measuring apparatus (see
fig 3). We can count the number N(a-up) of outcomes "a-up" or the number N(a-down) of
outcomes "a-down" and divide by the total number N of particles q that have participated in the
repeated measurements. If the relative frequencies ν(a-up) and ν(a-down) approximate real
numbers between 0 and 1 if N goes to infinity, then we call these real numbers the probabilities
P(a-up) and P(a-down). We can introduce the following probabilities : P( a, b) = the probability
that if particle q is in state of direction b, and the measurement m(a) is performed, the outcome "aup" will occur. P( -a, b) = the probability that if particle q is in state of direction b, and the
measurement m(a) is performed, the outcome "a-down" will occur.
To determine these probabilities, we should go to a laboratory and perform such repeated experiments, and then see what we find for the relative frequencies. We can also work out the
'mechanistic model' and then use our knowledge of classical mechanics to calculate the
probabilities. Let us regard the measurement situation of our classical macroscopical example a
little bit closer, and see which model we can propose. The three particles are located in a plane,
particle q in a point indicated by the direction b, and particles q
1
and q
2
in diametrically opposed
points indicated by the directions a and -a (see fig 4). Let us call γ the angle between
q
1
F
γ
q
γ/2
q
1
F
2
We consider the three particles from fig 3 as they are located
in a plane, particle q is located in the point indicated by the
direction b, particle q
1
and q
2
of the measuring apparatus are
located in the points indicated by the directions a and -a.
F
1
and F
and q
2
2
are the forces of attraction between q
and q (fig 4).
1
and q
2
the two space directions a and b. We propose the following mechanistic model :
1) The forces F1 and F2 of attraction are proportional to the product of the parameters
characterizing the particles divided by the square of the distance between the particles (these would
10
be Coulomb forces if q, q , and q
q , and q
1
2
1
2
represent charges, like in ref 16, and gravitational forces if q,
represent masses as in ref 17). Hence introducing a constant C we have :
|F | = C·
1
q ·q
1
2
2
l sin (γ/2)
|F | = C·
2
q ·q
2
2
(11)
2
l cos (γ/2)
2) The particle q moves under influence of the two forces F
1
and F
2
of the measurement ap-
paratus, and finally will arrive at rest at one of the two places indicated by direction a or -a,
depending on the magnitude of the forces of attraction between the three charges. If |F1| is bigger
than |F2| particle q will move, and arrive at the place of particle q . If |F1| is smaller than |F2|
particle q will move, and arrive at the place of particle q
2
1
.
3) The parameters q1 and q2 are arbitrary, their only constraint being that their sum must equal the
fixed parameter Q. This situation can be modelled by supposing that q1 is an at random number in
the interval [0,Q], and q2 = Q - q1. By means of these hypothesis 1, 2 and 3, we can make a
mathematical derivation for the probabilities P(a, b) :
P(a, b) = Probability that |F | is bigger than |F |
1

= P  C·

2
q ·q
1
2
2
l sin (γ/2)
2
> C·
q ·q


2
2

l cos (γ/2)
2
2
= P (q cos (γ/2) > q sin (γ/2))
1
2
2
2
= P (q cos (γ/2) > (Q − q ) sin (γ/2))
1
2
2
=
Q − Q·sin (γ/2)
Q
2
= cos (γ/2).
= P (q
1
2
> Q sin (γ/2))
(12)
This is exactly the probability that we would find if m(a) represented the measurement of the
spin of a spin 1/2 particle in the a direction while the particle has spin in the b direction.
We can describe this macroscopical system by the formalism of quantum mechanics. Every
state of the particle q in the direction b represented by spherical coordinates (θ, φ) is represented by
the unit vector
u
b
= (e
−iφ/2
·cos (θ/2 ), e
iφ/2
·sin (θ/2 )) with b = (cosφ·sinθ,sinφ·sinθ,cosθ ) (13)
of a two dimensional complex vector space as is well known for the spin of a spin 1/2 particle.
And the measurement m(a) where a = (cosβ·sinα, sinβ·sinα, cosα) is represented by means of the
self-adjoint operator
11
S
αβ
=

cosα

e


iβ
e
−iβ
·sinα
·sinα
−cosα



(14)

The eigenvalue +1 corresponds to the outcome "a-up" of the measurement m(a) and the eigenvalue
-1 to the outcome "a-down".We remark again that the state of the particle q is a pure state and the
probability only comes from a lack of knowledge about the measurement, or with other words, the
hidden variables are in the measurement, and not in the state of the system. These hidden variables
are the values of the parameters q1 and q2.We want to remark that other hypothesis about the
details of the mechanistic model can lead to other types of non-classical probability models, that in
certain occasions can be shown to be also non-quantum probability models (see ref 16 for details
on this aspect). In this paper, we only want to use this spin model for the construction of a
macroscopical classical entity that imitates the violation of the Bell inequalities of the quantum
entity consisting of two particles of spin 1/2 in the singlet spin state. So we are not really interested
how, and by which forces the classical motion is governed, as long as the model that we propose
remains purely mechanistic.
5.
The classical macroscopical example that imitates the quantum
mechanical violation of the Bell inequalities.
We shall now construct our classical macroscopical entity that violates Bell inequalities in
exactly the same way as does the quantum mechanical entity of the two spin 1/2 particles in the
singlet spin state.
The entity consists of two particles characterized by parameters q and s ( as we remarked in
the foregoing section we could take two positive charges, or two masses, for these two
parameters) that are bound on a rigid rod along the direction x, and move on this rigid rod in the
following way. At time t = 0 the two particles are in the centre c of the rod, and they move
outwards, q to the left and s to the right (see fig 5).
The coincidence measurements are constructed on the hand of analogue measurements than
the measurements m(a) introduced in the foregoing section. This means that at the two ends of the
rigid rod, we have measuring apparatuses constructed as described in the foregoing section. And
the rigid rods used for the construction of the measuring apparatuses are in a plane orthogonal to
the direction x (see fig 6).
12
centre c of the rod
The system of two particles q and s
direction x
q(t)
q(0) s(0)
s(t)
bounded on a rigid rod in direction x,
that move outwards, q to the left
and s to the right (fig 5).
Let us call A the circle with radius l/2, the collection of points where the particles q(1, a) and q(2,
a) of the left measuring apparatus can be located, and B the circle with radius l/2, the collection of
points where the particles s(1, b) and s(2, b) of the right measuring apparatus can be located. For
the measurement apparatus to perform measurement m(a) at the left we choose at random in an
interval [0,Q] the parameters q(1,a) and q(2,a) such that
q(1,a) + q(2,a) = Q, and locate them on
the rigid rod of the measurement apparatus, this means on the circle A, in the points a and -a.
direction a
F(1,a)
q(t)
q(2,a)
direction -a
F(2,a)
q(1,a)
direction x
The system of two particles q and s
direction b
q(0) s(0)
s(1,b)
that move outwards on a rigid rod
in direction x, together with the
F(1,b)
measuring apparatuses, that consists
of diametrically located particles
q(1,a) and q(2,a) on a circle A at
s(t)
F(2,b)
s(2,b)
direction -b
the left in a plane orthogonal to
x, and diametrically located particles
s(1,b) and s(2,b) on a circle B at the
right, in a plane orthogional to x (fig 6).
For the measurement apparatus to perform the measurement m(b) at the right, we choose at
random in an interval [0,S] the parameters s(1,b) and s(2,b) such that
s(1,b) + s(2,b) = S, and
locate them on the rigid rod of the measurement apparatus, this means on the circle B, in the points
b and -b. The two measurement apparatuses are at equal distances of the centre c of the rigid rod
on direction x, the place where the two particles q and s were at time t = 0. At time t the two
particles q and s have arrived at the centres of the two circles A and B.
If the two particles have arrived at the centres of the two circles, we can consider the
following forces. For the measurement apparatus at the left we consider two forces F(1,a), the
force between q and q(1,a), and F(2,a) the force between q and q(2,a). For the measurement
apparatus at the right we consider two forces F(1,b), the force between s and s(1,b), and F(2,b)
the force between s and s(2,b) (see fig 6). In this sense once the particles q and s have arrived at
the centres of the circles A and B, there are four forces F(1,a), F(2,a), F(1,b) and F(2,b) that are
considered. We now perform the measurement m(a, b) which is the performance of m(a) and
m(b) together.We propose the the following mechanistic model for this measurement :
One of the four forces is the biggest one in magnitude. Suppose that this biggest force is at the left,
hence it is F(1,a) or F(2,a). If F(1,a) is bigger than F(2,a), the particle q will move from the centre
of the circle A towards the particle q(1,a), and finally remain at rest in the point a of the circle A. If
13
F(1,a) is smaller than F(2,a), the particle q will move from the centre of the circle A towards the
particle q(2,a), and finally remain at rest in the point -a of the circle A. We suppose that the rigid
rod that connects the two particles q and s, can rotate freely around its centre c. Hence by the
motion of particle q, the motion of the particle s will be determined as well. If q arrives at the point
a of circle A, s will arrive at the point -a of the circle B. If q arrives at the point -a of the circle A,
s will arrive at the point a of the circle B. Because the particles q(1,a) and q(2,a) are chosen at
random in the interval [0,Q] we obviously have probability 1/2 that F(1,a) is bigger than F(2,a)
and hence the particles q and s move as shown in fig 7, and probability 1/2 that F(1,a) is smaller
than F(2,a) and hence the particles q and s move as shown in fig 8.
Till this moment the particles q and s have been connected by being attached to the rigid rod.
We make now the hypothesis that once the particles q and s touch the circles A and B, the connection breaks down. The particle q will not move any further, since it is already at point a or at
q
s(1,b)
direction a
If the force F(1,a) is bigger than
the force F(2,a) the particle q
will move towards the particle q(1,a)
and arrive finally at the point a of
the circle A. The particle s that is
F1
s
direction -a
q(2,a)
F2
still connected with q by means of
the rigid rod, will arrive at the point
-a of the circle B (fig 7).
s(2,b)
point -a of circle A.The particle s will however move further on, since it is still worked on by the
force F1 between s and s(1,b) and the force F2 between s and s(2,b) (see fig 7,8).
We are in a situation that has been treated in detail in the foregoing section and make the
same hypothesis than the one we have made in foregoing section for the nature of the mechanistic
evolution that follows. If F1 is bigger than F2 the particle s will move towards s(1,b), and finally
arrive in the point b of circle B. If F1 is smaller than F2 the particle s will move towards s(2,b),
and finally arrive in the point -b of circle B. The corresponding probabilities with which these
2
2
events happen are cos (γ/2) and sin (γ/2) where γ is the angle between the point where s is located
on the circle B and the point where s(1,b) is located as can be calculated (see 12).
If the biggest of the four forces is at the right, the symmetric mechanical motions as the ones
just explained are supposed to happen.
We have now described a measurement m(a, b) of which the possible outcomes are the
following (see fig 6) :
1) Outcome (a,b) which means that q arrives at a, and s arrives at b.
2) outcome (a,-b) which means that q arrives at a, and s arrives at -b.
3) outcome (-a,b) which means that q arrives at -a, and s arrives at b.
14
4) and outcome (-a,-b) which means that q arrives at -a and s arrives at -b.
q(1,a)
s(1,b)
If the force F(1,a) is smaller than
s
F1
the force F(2,a) the particle q will
move towards the particle q(2,a)
and arrive finally at the point -a
F2
of the circle A. The particle s that
is still connected with q by means
of the rigid rod, will arrive at the
point a of the circle B (fig 8).
s(2,b)
q
We will now calculate the probabilities for the measurement m(a, b) to give the different
possible outcomes.
Let us calculate the probability P(a,b) for outcome (a,b) to occur. In the case that the biggest
of the four forces is at the left, we have probability 1/2 that q arrives at point a of circle A, and
hence s arrives at point -a of circle B and then probability cos (γ/2) that s arrives at point b of
2
circle B where γ is the angle between -a and b. Hence the probability that the measurement m(a,
b) gives outcome (a,b) which means particle q at point a of circle A and particle s at point b of
circle B, is given by 1/2·cos (γ/2) where γ is the angle between -a and b. If we denote the angle
2
between a and b by arc(a,b), and remark that arc(a,b) + γ = π then we find :
2
P(a,b) = 1/2·sin (arc(a,b)/2)
(15)
And in a similar way we find :
2
P(a,-b) = 1/2·cos (arc(a,b)/2)
2
P(-a,b) = 1/2·cos (arc(a,b)/2)
2
P(-a,-b) = 1/2·sin (arc(a,b)/2)
(16)
In the case that the biggest of the four forces is at the left, from reasons of symmetry immediately
follows that we will find the same probabilities P(a,b), P(a,-b), P(-a,b) and P(-a,-b). If we
construct now the random variable corresponding to the measurement m(a, b), that consists of
giving value +1, if we have an outcome (a,b) or (-a,-b), and value -1, if we have outcome (a,-b)
or (-a,b), then the expectation value E(a,b) of this random value can easily be calculated.
E (a,b ) = (+1 ) ·{ P (a,b ) + P(−a,−b )} + (−1 )·{ P (a,−b ) + P(−a,b )}
= sin
2

arc(a,b) 
2

− cos
2

arc(a,b) 
2

= − cos(arc(a,b))
=
− a·b
(17)
15
This is exactly the same expectation value than the one calculated for the random variable introduced by Bell in relation with the coincidence spin measurements on the entity consisting of two
spin 1/2 particles in the singlet spin state, as we can see in (3).
If we now make the same choices for a, b, a', and b' as proposed in fig 1, then the expectation values E(a,b), E(a, b'), E(a',b), and E(a', b') corresponding to the measurements m(a, b),
m(a, b'), m(a', b), and m(a', b') will violate the Bell inequalities as calculated in (4).
A remark must be made :
As we have described our classical macroscopical measurement m(a, b), one could say that
it is not exactly a coincidence measurement, since the particle q at the left (or the particle s at the
right, depending on which of the four forces is the biggest one) reaches the point a or -a (or the
point b or -b), and then the particle s at the right still has to start moving towards the point b or the
point -b (or the particle q at the left still has to start moving towards the point a or -a). This is true.
The following answers can be given. For reasons of symmetry, it is obvious that a real
measurement of the type that we have proposed, will lead to coincidence outcomes. But the simple
mechanical model that we have proposed looses in a certain sense this symmetry. We could
consider a more complicated mechanistic model, that does not loose this symmetry, by considering
the motion of the particles q and s, taking into account the four forces F(1,a), F(2,a), F(1,b, and
F(2,b) at once. If we do this we find however a very complicated problem of classical mechanics,
that in principle is solvable, but not in an easy way. Bell inequalities will still be violated. We have
however introduced our simplification also for another reason. As we shall see, it is this
simplification that is very close to the 'algebraic calculations' on the quantum mechanical situation.
6.
Discussion of the macroscopical entity and further analogy with the
6.1.
If we compare the two examples, the classical macroscopical example with the vessels of
quantum case.
water, and the classical macroscopical example of the particles moving outwards on the rigid rod,
we see that the reason why in the case of the vessels of water we find a stronger violation of the
Bell inequalities, is due to the fact that this example is completely deterministic. It are the hidden
variables present in the measuring apparatus that 'soften' in a certain sense the violation of the Bell
inequalities in the example of section 5. Hence paradoxically, one can violate Bell inequalities
more severely by means of purely deterministic classical entities that form a whole and are then
broken apart, than by means of entities that form a whole and then are broken apart and are not
deterministic. The presence of quantum-like probabilities 'softens' the violation.
16
6.2
The detailed analysis that we have presented in section 3 remains completely valid for the
example of section 5. Indeed also here the violation is produced by the presence of what we have
called correlations of the second kind. It is the possibility of the rigid rod that connects
particles q and s, to rotate around its centre c, that correspond to the possibility of the quantum
particle to have a spin. The direction in which this rigid rod rotates correspond to the direction of
spin. And indeed, at the moment that the two particles q and s reach the centres of the circles A and
B, the rigid rod that connects them has not yet rotated around its centre in any direction. Only at
this moment the property that correspond in our model to the spin, is starting to get created. Since
the two particles q and s are still forming one entity, because connected by the rod, the 'joint
creation' of this property 'spin' will give rise to correlations of the second kind that typically
violate Bell inequalities.
6.3
Let us regard a little bit more in detail the calculation in quantum mechanics of the proba-
bilities P(a,b) :
P (a,b) = < Ψ
S
| P
a
⊗ P
b
| Ψ
S
(20)
>
where Pa and Pb are the projections on the vectors ua and ub of the two dimensional complex
Hilbert space that represents the spin states. If we take into account (2) and the fact that ΨS does
not depend on the direction n, we have :
P
a
⊗ P
b
( Ψ
S
) =
=
=
1
√2
1
√2
1
√2
(P
(u
a
a
⊗ P ) ·( u ⊗ u
b
⊗ P (u
·sin
b
a
−a
2
− u
−a
⊗ u
a
)
))
(
)
 arc a,b 

−a

(u
a
⊗ u
b
)
(19)
and since in general for any projector P in a Hilbert space, and any unit state vector Ψ of this
Hilbert space we have
< Ψ | P | Ψ > = < P ( Ψ ) | P( Ψ ) >
(20)
we can write
We present this calculation because in the algebraic steps of the calculation we can see a
similarity with the 'real' happenings in our macroscopical example. First the superposition singlet
state is reduced to a product state by the projection operators, and then the squares of the corresponding projection distances deliver the probabilities.
17
< Ψ
S
| P
a
⊗ P
b
| Ψ
S
> = < P
=
1
2
=
a
⊗ P ( Ψ
·sin
1
2
b
2
·sin

2

S
) | P
arc(a,b) 
2

a
⊗ P ( Ψ
· <u
b
a
⊗ u
b
S
) >
| u
a
⊗ u
b
>
arc(a,b) 
2

(21)
Let us now try to see that the 'description' of quantum mechanics 'indicates' that we are in a
situation of correlations of the second kind. This we can see by analyzing the form of the singlet
spin state of the entity consisting of two particles. We already remarked that this state as presented
in (2) is independent of the direction of the vector n. But there is more. In fact only one such state
can be made. Indeed also for different directions a, b in space we have that :
Ψ
S
= k (u
a
⊗ u
b
− u
b
⊗ u
a
) for some complex number k
(22)
This shows that ΨS does not depend on any direction of any vector in space. It is just a mathematical construction, using these vectors, that lead to a unique vector in the tensor product Hilbert
space. Hence ΨS does not represent a state of two particles which have already their spin,
although mathematically it is constructed in this way. It represents a state of an entity, consisting
of two particles, which do not have yet their spins. And the spins are created by the coincidence
measurement, that is exactly the measurement that takes apart the one entity, and breaks it into to
two 'separated' entities. In our example the situation is exactly the same. The state of the two
particles q and s when they are in the centres of the two circles A and B is such a 'singlet' state.
The rigid rod has no rotation around its centre s till this moment. The rotation is created from that
moment on.
7.
Is the concept of 'entities moving around and interacting inside space' at
stake ?
The two classical examples that violate Bell inequalities, that we have presented here, are of
course not 'physical models' for the real happening of the violation of the inequalities with the spin
entity. But they are in a certain sense 'philosophical models'. We can see by means of these
examples that there is something to explain in relation with the violation of Bell inequalities. The
only possible conclusion, compatible with a philosophical attitude of realism, seems to be to accept
that the entity of two quantum particles of spin 1/2 in the singlet spin state is an entity of nonseparated entities, till the moment that the coincidence measurement starts to 'separate' it.
18
This would mean that this one entity is present in a part of space of a macroscopical magnitude (of the order of 12 meters of length in the case of Aspects experiments). Is it imaginable that
two quantum-entities can form one whole in such a huge region of space? For the water, there is
no problem, because we can put a connecting tube, as long as we want, and for the charges we
can put a connecting rigid rod as long as we want, but for two quantum-entities?
Moreover, for example in the Aspect measurement, the space like parts of the photons, if
described by wave-packets, seem to fly apart, because they pass through two filters of different
frequencies. Hence, although it is very complicated to make a rigourous description of these space
like parts, they seem to be 'separated' in a certain sense. While all the 'non-local' aspects are due
to the spin-like parts of the entities. From this follows that we should not only 'imagine' ourselves
a photon-pair with the dimensions of a 'cloud' of 12 meter diameter, but with the additional fact,
that there is 'nothing' (we have to say 'nothing' in ordinary space) between. The particles remain
'one whole' while the space regions with probabilities of detecting one of the particles almost
equal to 1 get separated at macroscopical distances. This is certainly a situation that we will not be
able to imitate by means of a classical macroscopical entity.
Indeed, if we consider two space regions R1 and R2 that are macroscopicaly separated and
an entity S that constitutes a whole such that in the region R1 we have a probability almost equal to
1 to detect one part S1 (or S2) of the entity S, and in the region R2 we have a probability almost
equal to 1 to detect the other part S2 (or S1 ) of the entity S. And such that between the two regions
we have a probability almost equal to 0, to detect the entity, then this situation can only be realized
by means of a macroscopical entity when this macroscopical entity is already separated into two
sub entities. Macroscopical entities seem to have one additional property, that microscopical
entities not necessarily have. Let me call it the 'property of macroscopical wholeness'.
The property of macroscopical wholeness : For macroscopical entities we have the
following property : if they form a whole (hence are not two separated parts), then they hang together through space. Which means they cannot be localized in different macroscopicaly separated
regions R1 and R2 of space, without also being present in the region of space 'between' these
separated regions R1 and R2 .
To understand clearly this property of macroscopical wholeness, think of the example of the
water. If we cut of the connecting tube, the two vessels of water get localized in different separated
regions of space, but then they are separated entities, and Bell inequalities cannot be violated any
more with coincidence measurements on them. Microscopical entities seem to be able to constitute
a whole, without necessarily being submitted to this property of macroscopical wholeness. This is
in my opinion what we have to learn from the EPR experiments as a matter of fact.
19
If it were only the EPR measurements that cause us troubles of understanding about the
nature of reality, we could still hope to find a solution in the sense of questioning principles like
'causality, or Einstein locality etc..'. But other measurements, ever more strongly, and more
clearly, indicate that quantum-entities do not necessarily satisfy this property of macroscopical
wholeness. In fact it seems to be a very common thing for quantum entities. Together with J.
Reignier we are investigating the situation of a typical Stern-Gerlach measurement in relation with
this problematic
20,21)
. And we are trying to understand the reality of the state of such 'one'
particle, when it comes out of the Stern-Gerlach magnet. Our aim is to work out the meaning of
this state showing that this one particle (hence only one entity) effectively does not satisfy this
property of macroscopical wholeness. In a certain sense it is 'detectable' in two separated regions
R1 and R2 and not between.
It is very difficult for us to imagine that one entity as a whole does not satisfy this property
of macroscopical wholeness. But this difficulty probably comes exactly from the part of the reality
of our macroscopical space-concept, that we humans have constructed, from our experiences with
macroscopical entities that all do have this property. As we humans imagine space, and the entities
that are in it, we think that only two situations are possible.
Situation 1 : An entity forms a whole, and then breaking it into parts can make us perform
measurements that violate Bell inequalities, but then this entity must 'hang' together, and hence
cannot be localized in different separated regions of space.
Situation 2 : An entity is formed out of two separated entities, and then making experiments on the parts never will make it possible to violate Bell inequalities. In this case the separated
parts can of course be localized in different separated regions of space.
Other situations are very difficult for us to imagine. We repeat that in my opinion this comes
from the fact we have constructed space from our human experiences with macroscopical entities,
that exactly satisfy this macroscopical wholeness property.
We can speculate and invent a scenario of how the classical world-image of space filled up
with macroscopical entities that interact with each other 'inside' this space could have grown, the
macroscopical entities satisfying the macroscopical wholeness principle, although originally the
entities of which the macroscopical entities in it are constructed do not satisfy this macroscopical
13)
wholeness principle. A speculation of this kind is presented at the end of
. There will be needed
however a lot of investigation to be able to see in detail which aspects of the human construction of
the classical world-image with its macroscopical entities are to be relaxed to be able to 'understand'
the reality of the quantum world. An analysis of this kind is started in
22)
. Such an analysis does
not however have to start from nothing. We want to remark that already for more than thirty years
20
formalisms have been created with the main purpose of 'explaining' the quantum theory, but all
being founded on concepts more fundamental than the classical concepts of entities made up of
'substance' and interacting 'inside' a three dimensional Euclidean space. I myself have been
participating in the elaboration of one of these formalisms, originated by J.M. Jauch, founded by
C. Piron, and now commonly called the Geneva-formalism
23,24,25)
. In this type of formalisms it
must be possible to encounter and introduce the concepts necessary for the development of such a
physics, really detached from some of the old classical images, making impossible an
understanding of the quantum world till now. But a lot of work remains to be done.
To end this section we want to make one additional remark on this classical wholeness
principle, that is also in a subtle way connected to the problematic of non-locality touched upon in
this paper. The two aspects, the one of non-locality treated in this paper, and the one of
'incompleteness' of the quantum theory, both were presented in the original EPR paper. There is
the following fact : If one wants to describe a collection of quantum entities by the mathematical
formalism of ordinary quantum mechanics, then it can be shown, that such entities can never be
separated in the classical sense (hence situation 2 cannot be described). This is a shortcomings of
the mathematical structure (the vector space structure of its set of states, hence the superposition
principle) of the ordinary quantum formalism. By considering a more general formalism, as for
example the Geneva formalism , this shortcoming can be investigated in detail, as has been done in
7,24,26)
. As is shown in
7,24,26)
and more specifically in
27)
it is this shortcoming that is at the
origin of the 'logical' content of the EPR paper. An entity consisting of two separated quantum
entities cannot be described by the quantum formalism, since its collection of states cannot be
represented in a vector space. Hence what Einstein Podolsky and Rosen claim in their paper,
namely that the quantum formalism is 'incomplete' in the sense that it cannot represent all elements
of reality of an entity consisting of two separated quantum entities is correct (as explained by Max
Jammer in
19)
this is probably the part of the content of the EPR paper due to Boris Podolsky, and
his contacts with Kurt Gödel). In
27)
we point out explicitly the missing elements of reality. They
are however not missing states, as implicitly suggested by EPR and explicitly claimed by some
physicists who understood the EPR reasoning as a reasoning indicating a kind of incompleteness
(similar to the incompleteness of thermodynamics) that can be solved by adding hidden variables
to the description of the states, and in this way introducing more states. The incompleteness of
quantum theory is of a much more subtle nature, as is explained in
27)
.
8. Conclusion.
If one studies in detail the example of the mechanistic macroscopical laboratory situation,
violating the Bell inequalities 'in the same way as' the EPR experiments violations, than one can
21
notice that a lot of similar situations can be created. The necessary requirements to create such a
situation are :
To consider two entities that form a whole till a certain moment where they are separated by
the actions of the coincidence measurements. This separation produces two separated entities in a
'product' state of the type p ·p
a
one of the product states p ·p
a
one of the product states p
product state p ·p
a
b
, p
-a
·p
, or p
-a
·p
-b
-a
b
,or
·p
-a
or p ·p
b
a
a
-b
p
-b
·p
b
, p ·p
b
-b
such that at the right the realization of
happens with probability 1/2, or at left the realization of
happens with probability 1/2. The transition to the final
2
must then happen with a probability equal to the cos
of the half angle between
a,b. This is exactly a situation that we have created in section 5. But obviously it is possible to
invent other equivalent situations. It would be nice to construct along these lines a machine that
really could be built in the laboratory, to create this quantum-like violation of the Bell inequalities.
9.
References.
1) Bell, J., Physics 1,195 (1964).
2) Clauser, J.F. and Shimony, A., Reports on Progress in Physics 41, 1881 (1978).
3) d'Espagnat, B."A la Recherche du
Réel",
Gauthiers-Villars, Paris (1979).
4) Bell, J., "Bertlmann's socks and the nature of reality", in "Les implications Conceptuelles de la
Physique Quantique", Journal De Physique, 42
(1981).
5) See the articles in "Symposium on the Foundations of Modern Physics", eds.Lathi P. and
Mittelstaedt P., World Scientific, Singapore (1985), and in "Microphysical Reality and Quantum
Formalism", eds. van der Merwe A. et al, Kluwer Academic Publishers (1988).
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Physics Today, 39, 13 (1986).
7) Aerts, D., "The One and the Many", Doctoral thesis, TENA, Vrije Universiteit Brussel (1981).
8) Aerts, D., Lett. Nuovo Cim. 34, 107 (1982).
9) Aerts, D., " How do we have to change quantum mechanics in order to describe separated
systems? "
in "The Wave-Particle Dualism", eds. Diner S. et al., Reidel, Dordrecht (1984).
10) Aerts, D., " The physical origin of the Einstein Podolsky Rosen paradox "
in "Open
Questions in Quantum Physics", eds. Tarozzi G. and van der Merwe A., Reidel, Dordrecht
(1985).
11) Aerts, D., "The physical origin of the EPR paradox and how to violate Bell inequalities by
macroscopic systems"
in "Symposium on the Foundations of Modern Physics", eds.Lathi P. and
Mittelstaedt P., World Scientific, Singapore (1985).
12) Aerts, D., "The description of separated systems and quantum mechanics and a possible
explanation for the probabilities of quantum mechanics"
in "Microphysical Reality and Quantum
Formalism", eds. van der Merwe A. et al, Kluwer Academic Publishers (1988).
22
13) Aerts, D., "An attempt to imagine parts of the reality of the micro-world", to appear in the
proceedings of the conference "Problems in Quantum Physics ; Gdansk '89'" to be published by
World Scientific Publishing Company, Singapore (1990).
14) Bell, J., "Foundations of Quantum Mechanics",
(1971).
ed. B. d'Espagnat, New-York; Academic
15) Bohm, D., "Quantum Theory", Prentice-Hall (1951).
16) Aerts, D.,
J. Math. Phys. 27 , 202 (1986).
17) Aerts, D., "The origin of the non-classical character of the quantum probability model", in "
Information, complexity, and control in quantum physics", edited by A. Blaquiere, S. Diner, and
G. Lochak, Springer-Verlag, Wien-New York, (1987).
18) Aerts, D., "A possible explanation for the probabilities of quantum mechanics and a
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